Theorem funco 4457

Description: The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (The proof was shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
funco	|- ((Fun F /\ Fun G) -> Fun (F o. G))

Proof of Theorem funco

Step	Hyp	Ref	Expression
1		moexexv 1842	. . . . . 6 |- ((E*z xGz /\ A.zE*y zFy) -> E*yE.z(xGz /\ zFy))
2	1	ancoms 484	. . . . 5 |- ((A.zE*y zFy /\ E*z xGz) -> E*yE.z(xGz /\ zFy))
3		funmo 4437	. . . . . 6 |- (Fun F -> E*y zFy)
4	3	19.21aiv 1664	. . . . 5 |- (Fun F -> A.zE*y zFy)
5		funmo 4437	. . . . 5 |- (Fun G -> E*z xGz)
6	2, 4, 5	syl2an 503	. . . 4 |- ((Fun F /\ Fun G) -> E*yE.z(xGz /\ zFy))
7	6	19.21aiv 1664	. . 3 |- ((Fun F /\ Fun G) -> A.xE*yE.z(xGz /\ zFy))
8		funopab 4455	. . 3 |- (Fun {<.x, y>. | E.z(xGz /\ zFy)} <-> A.xE*yE.z(xGz /\ zFy))
9	7, 8	sylibr 217	. 2 |- ((Fun F /\ Fun G) -> Fun {<.x, y>. | E.z(xGz /\ zFy)})
10		df-co 4003	. . 3 |- (F o. G) = {<.x, y>. | E.z(xGz /\ zFy)}
11	10	funeqi 4442	. 2 |- (Fun (F o. G) <-> Fun {<.x, y>. | E.z(xGz /\ zFy)})
12	9, 11	sylibr 217	1 |- ((Fun F /\ Fun G) -> Fun (F o. G))

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296  E.wex 1326  E*wmo 1772   class class class wbr 3338  {copab 3395   o. ccom 3990  Fun wfun 3992

This theorem is referenced by:  fnco 4521  fcoOLD 4574  f1co 4612  fvco 4736  curry1 5075  curry2 5078  mapenlem1 5583  vsfval 9586  domrancur1b 14548  domrancur1c 14550  f1ocan1fv 15717

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-fun 4008

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